This kind of "awkwardness" is felt as a result of being stuck in your head. It's only awkward if you're worried about making a mistake. But you can relax, you're allowed to make mistakes. No one except you really cares if you said something stupid or rude or pointless.
No one except you really cares if you said something [..] rude
I don't think that's true and I think that it shows that you have never actually broken off an unwanted conversation by being honest. If you do that and, for instance, say "I'm sorry, but this conversation really doesn't interest me", people will feel insulted. You are forced to beat around the bush. [or] stupid or pointless
The fact that you yourself care about those is enough. You don't want to feel forced to say stupid or pointless things.There are two ways out:
1) Be interested in what people have to say about certain subjects and steer the conversation there. Those subjects have to be general enough that everyone has something to say about them. Sometimes you turn out to be annoying, for instance when you are cross-examining someone about why they are happy with their lives.
2) Accept the fact that people will think you're weird or an asshole. Me, I just stay silent when there's nothing left to say. There's nothing wrong with silence. When the silence lasts long enough, I turn around and walk.